To factor the quadratic expression [tex]\( x^2 - 4x - 5 \)[/tex], we need to express it as a product of two binomials. Let's go through the steps:
1. Identify the quadratic expression: We start with [tex]\( x^2 - 4x - 5 \)[/tex].
2. Set up the binomials: We will write the quadratic expression in the form [tex]\( (x + a)(x + b) \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants to be determined.
3. Expand the binomials: When expanded, the expression [tex]\( (x + a)(x + b) \)[/tex] forms [tex]\( x^2 + (a + b)x + ab \)[/tex].
4. Match coefficients: Compare the expanded form [tex]\( x^2 + (a + b)x + ab \)[/tex] to the original quadratic expression [tex]\( x^2 - 4x - 5 \)[/tex]. We see that:
- The coefficient of [tex]\( x \)[/tex] is [tex]\(-4\)[/tex], so [tex]\( a + b = -4 \)[/tex].
- The constant term is [tex]\(-5\)[/tex], so [tex]\( ab = -5 \)[/tex].
5. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]: We need two numbers that add up to [tex]\(-4\)[/tex] and multiply to [tex]\(-5\)[/tex]. These numbers are [tex]\( -5 \)[/tex] and [tex]\( 1 \)[/tex], because:
- [tex]\(-5 + 1 = -4\)[/tex]
- [tex]\(-5 \times 1 = -5\)[/tex]
6. Write the factored form: Thus, the quadratic expression [tex]\( x^2 - 4x - 5 \)[/tex] factors as [tex]\( (x - 5)(x + 1) \)[/tex].
Therefore, the factored form of [tex]\( x^2 - 4x - 5 \)[/tex] is [tex]\( (x - 5)(x + 1) \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{(x - 5)(x + 1)} \][/tex]
